This thesis approaches systemic risk from an epidemiology perspective, modeling the
transfer of disease as a dynamical system and attempting to quantify the risk of
asymptotic instability. This instability specifically means trending to a probability
of infection for each member of the system that is nonzero. In this work, we (1)
extend a susceptible-infected-susceptible model for epidemic spreading to allow for
asymmetry and time-variance, (2) bound the epidemic threshold of the model with
the joint spectral radius (JSR) of the relevant transition matrices, and (3) use a
semide nite programming technique to compute an upper bound on the JSR. We
also experiment with two real disease models{HIV and Zica{to validate our model as
well as to test our the impact of the network parameters, transmission rates, and cure
rates on the epidemic threshold. Our results indicate that allowing time variance in
these asymmetric models requires a more complex computational tool for providing
an upper bound because the interaction between matrices can lead to higher JSR's
than repeating any individual transmission matrix would. Finally, we provide one
example of an interventionist use for this model by analyzing the effect of removing
one node from the graph and compare random selection vs. maximally connected
selection for that node.